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proof-sheets were read.* The Academy (in 1830) made amends
to Abel by awarding him the Grand Prize in Mathematics
jointly with Jacobi. Abel, however, was dead.
The opening paragraphs of the memoir indicate its scope.
The transcendental functions hitherto considered by
mathematicians are very few in number. Practically the
entire theory of transcendental functions is reduced to
that of logarithmic functions, circular and exponential
functions, functions which, at bottom, form but a single
species. It is only recently that some other functions have
begun to be considered. Among the latter, the elliptic
transcendents, several of whose remarkable and elegant
properties have been developed by Mr Legendre, hold the
first place. The author [Abel] has considered, in the memoir
which he has the honour to present to the Academy, a very-
extended class of functions, namely: all those whose deri-
vatives are expressible by means of algebraic equations
whose coefficients are rational functions of one variable,
and he has proved for these functions properties analogous
to those of logarithmic and elliptic functions ... and he
has arrived at the following theorem:
If we have several functions whose derivatives can be
roots of one and the same algebraic equation, all of whose
coefficients are rational functions of one variable, we can
always express the sum of any number of such functions
by an algebraic and logarithmic function, provided that we
establish a certain number of algebraic relations between
the variables of the functions in question.
The number of these relations does not depend at all
upon the number of functions, but only upon the nature of
the particular functions considered. ...
* Libri, a soi-disant mathematician, who saw the work through the
press, adds, *by permission of the Academy*, a smug footnote acknow-
ledging the genius of the lamented AbeL This is the last straw; the
Academy might have come out with all the facts or have held its
official tongue. But at all costs the honour and dignity of a stuffed
shirt must be upheld. Finally it may be recalled that valuable manu-
scripts and books had an unaccountable trick of vanishing when Libri
was round.